Squares Definition and 383 Threads

Here is the question from the book: ------ Let n\geq1 and let a_1,...,a_n and b_1,...,b_n be real numbers. Verify the identity: \left(\sum_{i=1}^n{a_ib_i}\right)^2 + \frac{1}{2}\sum_{i=1}^n{\sum_{j=1}^n{\left(a_ib_j-a_jb_i\right)^2}} =...

Homework Statement X mod m is the remainder when x is divided by m. This value is called a residue. Find all perfect squares from the set of residues mod 16. The Attempt at a Solution There was a suggestion that this would become clearer when the definition of perfect square was...

Hi all, I think this sounds like a really simple and trivial question, but I've no clue as to where i should start: true or false? between any two different positive rational numbers lies the square of a rational number. while i can't provide a construction of such a number, i somehow...

I was reading through a proof of the summation formula for a sequence of consecutive squares (12 22 + 32 + ... + n2), and the beginning of the proof states that we should take the formula: (k+1)3 = k3 + 3k2 + 3k + 1 And take "k = 1,2,3,...,n-1, n" to get n formulas which can then be...

Hello i need help with a question, other people tried to help me, i just cannot get it! its driving me crazy Two positive numbers have sum n. What is the smallest value possible for the sum of their squares? so i have n=x+y x>0 y>0 y=n-x we want to minimize S S=x^2+y^2...

Which squares are expressible as the sum of two squares? Is there a simple expression I can write down that will give me all of them? Some of them? Parametrization of the pythagorean triples doesn't seem to help.

Suppose X is a set consisting of squares with the property that any addition with elements of X (where no two are the same) gives a square (might not be in X). How many elements can X have?

Greetings, I'm curious if "magic squares" have been found to be useful in mathematical or scientific endeavors apart from an "oddity" or "game"

This has to do with the 2nd order recursive sequence \{...a, b, c ...\} where a,b,c are any three sucessive terms and c = 6b-a + 2k. I found that it has the following property. 8ab - (a+b-k)^2 = 8bc - (b+c-k) That is eight times the product of two adjacent terms always equals the square of...

How do you go from \sum_{n = 1}^n i^2 to \frac{n(n + 1)(2n + 1)}{6}?

We all know the least squares method to find the best fit line for a collection of random data. But I wonder if it is the best method. Suppose we have two random variables y and x that appear to have a linear relation of the type y = ax+b. What we want is, given the next type x signal to...

Hi, I was working on a problem and I can't figure out what I'm supposed to do. It reads, find the vector in subspace S that is closest to v; write v as the sum of a vector in S and a vector in S^a; and find the distance from v to S. S spanned by {(1,3,4)} v = (2,-5,1) Ok, what I did was...

Question states Consider the vector space C[-1,1] with an inner product defined by <f,g> = the integral from 1 to -1 of f(x)g(x) dx a) Show that u1(x)= 1/(2^.5) u2(x)= ((6^.5)/2)x form an orthonormal set of vectors b) Use the result from a) to find the best least squates...

a number n when factorised can be written as a^4*b^3*c^7.find number of perfect square which are factors of n.a,b,c are prime >2. I have no idea how to start? please help.

A figure contains five equal squares in the form of a cross. Can you show how to divide this figure into four equal parts which will fit together to form a square

It seems to me that Linear Regression and Linear Least Squares are often used interchangeably, but I believe there to be subtle differences between the two. From what I can tell (for simplicity let's assume the uncertainity is in y only), Linear Regression refers to the general case of fitting...

ok, i need to derive a forumla that will add the consecutive squares of n numbers. for example 1^2 + 2^2 + 3^2 + ... + (n-2)^2 + (n-1)^2 + (n)^2 I have worked on this problem for quite some time and haven't been able to come up with anything. I do know that the sum of consecutive...

Prove that the square of an odd integer is always of the 8k + 1, where k is an integer. Any help would be appreciated.

Hi, I am having some difficulty with this problem: what would be Y^h^a^t if s_y_/_x = 439, n = 24 and 95% confidence interval estimate for the average Y given a particular value of X is 1125 and 1695. ----------------- I know Y^h^a^t = b_o + b_1x but I am not sure how I can use the...

Start with a 8 by 8 board of squares (for instance, a chessboard) and take away the top-left and bottom-right corner squares so that there are 62 squares left. Take some dominoes that are the same size as two squares of the chessboard. Can you cover the 62 squares of the board with 31 such...

How do I factor this equation x^3-1

Greetings friends, I have come across an argument on cancelling the squares on either side of an equation. For example if the equation is (a-b)^2=(c-b)^2 my argument is that i can cancel the squares by taking the square root of both sides as to get (a-b)=(c-b) and hence a=c. But others says...

\begin{array}{c} {{A_n}={\sqrt{\sum _{z=1}^{n}{z^2}}} } \\ {{A_1}=1 } \\ {{A_{24}}=70}\end{array}\ Is there a proof that only for n =1 or n=24 that An is an integer quantity?

Magic Square Hello, Don't know, which forum, so i put it to general... Yesterday i saw something like an magician on an exposition, showing some math to angle for attention. He asked the audience to give him a number between 41 and 100. So he got the 47. He worked out a magic...

How do I find out if a number is a perfect square modulo a? For example, is 4 (mod 10) a perfect square?

Compute the following: \sum_{n=1}^{+\infty} \frac{1}{n^{2}} =...?? \sum_{n=1}^{+\infty} \frac{1}{n^{4}} =...?? .LINKS TO WEBPAGES WITH SOLUTIONS ARE NOT ALLOWED! :-p Daniel.

Construct the normal equations for the linear polynomial least squares to fit the data x = [1 0 -1], y=[3;2;-1]. (a) Find the parameters of the linear regression u1, u2 using QR decomposition, and plot the data and the fit curve in a graph (paper and pencil). (b) Calculate the eigenvalues of the...

urgent! finding co-ordinates in squares A(7,2) and C(1,4) are vertices of a square ABCD. equation BD is y=3x-9 midpoints AC: (4,3) find the co-ordinates of B and D i just don't understan how to get the co-ordinates, I've tried plotting a graph but to no avail. i found the midpoints...

First I'd like to say that I'm getting back into college after several years out in the job market. Unfortunately, I need to complete several more upper division math courses before I can complete my CS degree. Before I go back and start taking my classes again, I've been trying to self-study...

A little while ago I noticed a pattern in the sums of the digits of perfect squares that seems to suggest that: For a natural number N, the digits of N^2 add up to either 1, 4, 7, or 9. ex: 5^2 = 25, 2+5 = 7 In some cases, the summation must be iterated several times: ex: 7^2 = 49...

I'm getting confused and can't seem to wrap my head around this problem. Prove that the sum of the squares of any 3 consecutive odd numbers when divided by 12 gives a remainder of 11. I'm not sure how to set this up or proceed I figured that (n^2 + (n +2)^2 + (n+4)^2)/12 = x + 11...

perfect square? hello everyone! i have come across an on-going research onDETERMINING A PERFECT SQUARE GIVEN A DIFFERENCE . However, I have a feeling that this was not an original one. the researcher used the Pythagorean theorem to arrive at his so called "theorem". would anyone give...

Greetings, I suppose all of us have at one time or another been fascinated by "magic squares" My question is: has the relationship of numbers in a magic square been found to be useful in the mathematical sciences in any advanced analytical work? Or is is just a mathematical curiousity?